What is the integral of (13)/(13+e^x) ?

The integral of (13)/(13+e^x) is x-ln|e^x+13|+C

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integral of (13)/(13+e^x)

\int \frac{13}{13 + e ^{x}} d x

x - ln ∣ e^{x} + 13 ∣ + C

Solution steps

\int \frac{13}{13 + e ^{x}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 13 \cdot \int \frac{1}{13 + e ^{x}} d x

Apply u-substitution

: \int \frac{1}{u ( 13 + u )} d u

= 13 \cdot \int \frac{1}{u ( 13 + u )} d u

Take the partial fraction of

\frac{1}{u ( 13 + u )} : \frac{1}{13 u} - \frac{1}{13 ( u + 13 )}

= 13 \cdot \int \frac{1}{13 u} - \frac{1}{13 ( u + 13 )} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 13 (\int \frac{1}{13 u} d u - \int \frac{1}{13 ( u + 13 )} d u)

\int \frac{1}{13 u} d u = \frac{1}{13} ln ∣ u ∣

\int \frac{1}{13 ( u + 13 )} d u = \frac{1}{13} ln ∣ u + 13 ∣

= 13 (\frac{1}{13} ln ∣ u ∣ - \frac{1}{13} ln ∣ u + 13 ∣)

Substitute back

u = e^{x}

= 13 (\frac{1}{13} ln ∣ e^{x} ∣ - \frac{1}{13} ln ∣ e^{x} + 13 ∣)

Simplify

13 (\frac{1}{13} ln ∣ e^{x} ∣ - \frac{1}{13} ln ∣ e^{x} + 13 ∣) : x - ln ∣ e^{x} + 13 ∣

= x - ln ∣ e^{x} + 13 ∣

Add a constant to the solution

= x - ln ∣ e^{x} + 13 ∣ + C

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\int e^{x} \cdot x^{- 1} d x

\frac{d y}{d x} = \frac{4 y ln ( x )}{x}

(1.23) (2.46)

\int_{\frac{π}{2}}^{π} 2 cot (\frac{x}{2}) d x

What is the integral of (13)/(13+e^x) ?
The integral of (13)/(13+e^x) is x-ln|e^x+13|+C